Nonlinear Analysis 51 (2002) 1363–1372 www.elsevier.com/locate/na Approximation of random xed points in normed spaces Ismat Beg Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan Received 24 June 2000; accepted 25 June 2001 Keywords: Random xed point; Nonexpansive random map; Banach space: Measurable space 1. Introduction Over the last 30 years, random operator theory has grown into a full edged research area and various ideas associated with random xed point theory are used to form a particularly elegant approach for the solution of nonlinear random systems (see [6]). Random xed point theorems for random contraction mappings on Polish spaces were rst proved by Spacek [21] and Hans [9,10]. Subsequently Bharucha-Reid [7] has given sucient conditions for a stochastic analogue of Schauder’s xed point theorem for a random operator. Itoh [12] introduced random condensing operators and considerably improved the known results. Sehgal and Water [19,20] considered Browder-Fan type random operators and as a consequence obtained a stochastic generalization of the well-known Rothe xed point theorem. Recently Papageorgiou [18], Xu [25], Beg [2,3], Tan and Yuan [22,23], Liu [15], Beg and Shahzad [4,5] and many other authors have studied the xed points of random maps. Mann [16], Outlaw [17], Ishikawa [11] and Ghosh and Debnath [8] had used dierent iteration schemes to obtain xed points in deterministic operator theory. The aim of this paper is to study the behaviour of the iterate of a nonexpansive random map on an arbitrary Banach space. We proved the existence of unique random xed points of nonexpansive random mappings in Banach spaces. Our results are noteworthy in the sense that no geometric assumption is required on the underlying space. E-mail address: ibeg@lums.edu.pk (I. Beg). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0362-546X(01)00902-6